Solve for $x$ : $ 7|x + 8| + 1 = -2|x + 8| + 2 $
Add $ {2|x + 8|} $ to both sides: $ \begin{eqnarray} 7|x + 8| + 1 &=& -2|x + 8| + 2 \\ \\ { + 2|x + 8|} && { + 2|x + 8|} \\ \\ 9|x + 8| + 1 &=& 2 \end{eqnarray} $ Subtract ${1}$ from both sides: $ \begin{eqnarray} 9|x + 8| + 1 &=& 2 \\ \\ { - 1} &=& { - 1} \\ \\ 9|x + 8| &=& 1 \end{eqnarray} $ Divide both sides by ${9}$ $ \dfrac{9|x + 8|} {{9}} = \dfrac{1} {{9}} $ Simplify: $ |x + 8| = \dfrac{1}{9}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 8 = -\dfrac{1}{9} $ or $ x + 8 = \dfrac{1}{9} $ Solve for the solution where $x + 8$ is negative: $ x + 8 = -\dfrac{1}{9} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& -\dfrac{1}{9} \\ \\ {- 8} && {- 8} \\ \\ x &=& -\dfrac{1}{9} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $9$ $ x = - \dfrac{1}{9} {- \dfrac{72}{9}} $ $ x = -\dfrac{73}{9} $ Then calculate the solution where $x + 8$ is positive: $ x + 8 = \dfrac{1}{9} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& \dfrac{1}{9} \\ \\ {- 8} && {- 8} \\ \\ x &=& \dfrac{1}{9} - 8 \end{eqnarray} $ Change the ${ - 8}$ to an equivalent fraction with a denominator of $9$ $ x = \dfrac{1}{9} {- \dfrac{72}{9}} $ $ x = -\dfrac{71}{9} $ Thus, the correct answer is $x = -\dfrac{73}{9} $ or $x = -\dfrac{71}{9} $.